Tbilisi Mathematical Journal

Existence of positive solutions of four-point BVPs for one-dimensional generalized Lane-Emden systems on whole line

Pinghua Yang and Yuji Liu

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Abstract

This paper is concerned with four-point boundary value problems of the one-dimensional generalized Lane-Emden systems on whole lines. The Green's functions $G(t,s)$ for the problem $-(\rho(t)x'(t))'=0$ with boundary conditions $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}x(t)-l x(\eta)=0$ and $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}\rho(t)x'(t)-l \rho(\eta)x'(\eta)=0$ are obtained respectively. We proved that $G(t,s)\ge 0$ under some assumptions which actually generalize a corresponding result in [J. Math. Anal. Appl. 305 (2005) 253-276]. Sufficient conditions to guarantee the existence of positive solutions of this kind of models are established. Examples are given at the end of the paper.

Article information

Source
Tbilisi Math. J., Volume 8, Issue 2 (2015), 257-280.

Dates
Received: 18 November 2014
Accepted: 21 October 2015
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528769021

Digital Object Identifier
doi:10.1515/tmj-2015-0026

Mathematical Reviews number (MathSciNet)
MR3441141

Zentralblatt MATH identifier
1336.34043

Subjects
Primary: 34B10: Nonlocal and multipoint boundary value problems
Secondary: 34B15: Nonlinear boundary value problems 35B10: Periodic solutions

Keywords
One-dimensional generalized Lane-Emden system four-point boundary value problem positive solution fixed point theorem

Citation

Yang, Pinghua; Liu, Yuji. Existence of positive solutions of four-point BVPs for one-dimensional generalized Lane-Emden systems on whole line. Tbilisi Math. J. 8 (2015), no. 2, 257--280. doi:10.1515/tmj-2015-0026. https://projecteuclid.org/euclid.tbilisi/1528769021


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